(Locally) shortest arcs of a special sub-Riemannian metric on the Lie group $\mathrm {SO}_0(2,1)$
HTML articles powered by AMS MathViewer
- by
V. N. BerestovskiÄ
Translated by: the author - St. Petersburg Math. J. 27 (2016), 1-14
- DOI: https://doi.org/10.1090/spmj/1373
- Published electronically: December 7, 2015
- PDF | Request permission
Abstract:
The geodesics, shortest arcs, cut loci, and conjugate sets are found for a left-invariant sub-Riemannian metric on the Lie group $\mathrm {SO}_0(2,1)$ under the condition that the metric is right-invariant relative to the Lie subgroup $\mathrm {SO}(2)\subset \mathrm {SO}_0(2,1)$.References
- Shigeo Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2) 10 (1958), 338ā€“354. MR 112152, DOI 10.2748/tmj/1178244668
- Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. BĆ©rard-Bergery, M. Berger and J. L. Kazdan. MR 496885
- V. N. Berestovskii and Luis Guijarro, A metric characterization of Riemannian submersions, Ann. Global Anal. Geom. 18 (2000), no.Ā 6, 577ā€“588. MR 1800594, DOI 10.1023/A:1006683922481
- V. N. BerestovskiÄ and I. A. Zubareva, Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups, Sibirsk. Mat. Zh. 42 (2001), no.Ā 4, 731ā€“748, i (Russian, with Russian summary); English transl., Siberian Math. J. 42 (2001), no.Ā 4, 613ā€“628. MR 1865469, DOI 10.1023/A:1010439312070
- V. N. BerestovskiÄ, Universal methods for the search for normal geodesics on Lie groups with a left-invariant sub-Riemannian metric, Sibirsk. Mat. Zh. 55 (2014), no.Ā 5, 959ā€“970 (Russian, with Russian summary); English transl., Sib. Math. J. 55 (2014), no.Ā 5, 783ā€“791. MR 3289106, DOI 10.1134/s0037446614050012
- A. V. Pogorelov, Differential geometry, P. Noordhoff N. V., Groningen, 1959. Translated from the first Russian ed. by L. F. Boron. MR 0114163
- Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
- F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Chelsea Publishing Co., New York, 1959. Translated by K. A. Hirsch. MR 0107649
- D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR 0229177
Bibliographic Information
- V. N. BerestovskiÄ
- Affiliation: Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences
- Email: vberestov@inbox.ru
- Received by editor(s): June 10, 2014
- Published electronically: December 7, 2015
- Additional Notes: Partially supported by RFBR (grant no.Ā 14-01-00068-p) and by a grant of the Government of the Russian Federation for the State Support of Scientific Research (agreement no. 14.B25.31.0029)
- © Copyright 2015 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 1-14
- MSC (2010): Primary 22E30, 49J15, 53C17
- DOI: https://doi.org/10.1090/spmj/1373
- MathSciNet review: 3443263